Problems

In Problems 1-36, test the improper integral for convergence and evaluate when possible.

37            Show that if r is a rational number, the improper integral converges when r < 1 and diverges when r > 1.

38            Show that if r is rational, the improper integral converges when r > 1 and diverges when r < 1.

39            Find the area of the region under the curve y = 4x^-2 from x = 1 to x = x.

40            Find the area of the region under the curve from x = ½ to x = 1.

41             Find the area of the region between the curves y = x^-1/4 and y = x^-1/2 from x = 0 to x = 1.

42            Find the area of the region between the curves y = -x^-3 and y = x^-2, 1 ≤ x < x.

43            Find the volume of the solid generated by rotating the curve y = l/x, 1 ≤ x < x, about (a) the x-axis, (b) the y-axis.

44            Find the volume of the solid generated by rotating the curve y = x^-1/3, 0 < x ≤ 1, about (a) the x-axis, (b) the y-axis.

45            Find the volume of the solid generated by rotating the curve y = x^-3/2, 0 < v ≤ 4, about (a) the x-axis, (b) the y-axis.

46            Find the volume generated by rotating the curve y = 4x^-3, -∞ < ∞ ≤ -2, about

(a) the x-axis, (b) the y-axis.

47            Find the length of the curve from x = 0 to x = 1.

48            Find the length of the curve y = ¾x^1/3 - 3/5 x^5/3 from x = 0 to x = 1.

49            Find the surface area generated when the curve y = √x - ⅓x√x, 0 ≤ x ≤ 1, is rotated about (a) the x-axis, (b) the y-axis.

50            Do the same for the curve y = ¾x^1/3 - 3/5 x^5/3,0 ≤ x ≤ 1.

51            (a) Find the surface area generated by rotating the curve y = √x, 0 ≤ x ≤ 1, about

the x-axis.

(b)  Set up an integral for the area generated about the y-axis.

52             Find the surface area generated by rotating the curve y = x^2/3, 0 ≤ x ≤ 8, about the x-axis.

53            Find the surface area generated by rotating the curve , 0 ≤ x ≤ a, about (a) the x-axis, (b) the y-axis (0 < a ≤ r).

54            The force of gravity between particles of mass m₁ and m₂ is F = gm₁m₂/s² where s is the distance between them. If m₁ is held fixed at the origin, find the work done in moving m₂ from the point (1,0) all the way out the x-axis.

H 55           Show that the Rectangle and Addition Properties hold for improper integrals.